We prove a monoidal equivalence, called universal Koszul duality, between genuine equivariant K-motives on a Kac–Moody flag variety and constructible monodromic sheaves on its Langlands dual. The equivalence is obtained by a Soergel-theoretic description of both sides which extends results for finite-dimensional flag varieties by Taylor and the first author. Universal Koszul duality bundles together a whole family of equivalences for each point of a maximal torus. At the identity, it recovers an ungraded version of Beilinson, Ginzburg and Soergel’s and Bezrukavnikov and Yun’s Koszul duality for equivariant and unipotently monodromic sheaves. It also generalizes Soergel-theoretic descriptions for monodromic categories on finite-dimensional flag varieties by Lusztig and Yun, Gouttard and the second author. For affine Kac–Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis and Kamnitzer as well as Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.

We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern–Simons functional and prove a localization result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else ‘escape to infinity’. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localizing critical points. As an application, we obtain formulas for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by 4) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold.

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups $H_1$ and $H_2$, we study the tensor product of the associated representation rings $\mathrm{R}H_1$ and $\mathrm{R}H_2$ over the representation ring $\mathrm{R}G$, and prove a vanishing result for the associated higher Tor groups. This result can be viewed as a natural extension of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over $\mathrm{R}G$. It may also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.

Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao (Growth tight actions, Pacific J. Math. 278 (2015), 1–49).

We establish several surprising rigidity results for étale groupoids associated with algebraic actions. Our initial motivation comes from algebraic number theory, where our results are witnessed in a particularly striking fashion. We prove that the groupoids associated with the action of the multiplicative monoid in a ring of algebraic integers on the additive group of the ring remember the ring up to isomorphism. Already, this special case resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch–Uchida theorem using topological full groups.